Start with an icosahedron. Inscribe within it a dodecahedron with a vertex centered on each face (dual polyhedron).
The "compound of five cubes" shows one can inscribe a cube within a dodecahedron by selecting 8 vertices. The remaining 12 dodecahedron vertices sit 2 per cube face, roughly evenly spaced on an orthogonal line through the center of each square. Build a d12 cubical die which has two pegs per face instead of pips. Each cube edge corresponds to a dodecahedron face.
A tetrahedron may be inscribed within a cube by selecting face diagonals. There are two ways: the union is a stella octangula; the intersection, a octahedron.
Inscribe an octahedron inside a tetrahedron by joining midpoints of the edges.
The octahedron has square cross sections, so this is how the Pyramorphix is equivalent to the Pocket Cube. Build a twisty puzzle (probably virtual) which twists both like a Pyramorphix and Pyraminx. Or a both face turning and vertex turning octahedron -- could be done with LEDs. Text and and static images do a poor job of describing twisty puzzles on Wikipedia; video is great. Slow motion, fast forward, zoom, brighten, loop segment are needed.
From a cube, one can get to all regular polyhedra. Isn't this beautiful?
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